Tree density estimation

A natural strategy for mitigating the curse of dimensionality in estimating probability distributions is to employ lowcomplexity family of approximation distributions. For discrete distributions, Chow and Liu [5] suggested a family of tree-based approximations and gave an efficient maximum-likelihood estimator based on Kruskal's optimal spanning tree algorithm [14]. We stress that this approach makes no structural assumptions about the sampling distribution, but rather constitutes a modeling choice. Consequently, in this paradigm, the goal is to approximate the optimal-tree distribution from the data, without any guarantees on how well the latter approximates the true sampling distribution. Extensions of the Chow-Liu approach to continuous distributions were studied by Bach and Jordan [1] and by Liu et al. [16] under various assumptions.